NSWC-11 Gear & Spline Reliability Model

A gear fails when it can no longer do its job efficiently, which is not a sharp line. Noise, vibration and inspection results decide it. Gears wear, pit, flow plastically or break, and well-made gears in a clean, correctly aligned mesh are very reliable. The reliability is mostly lost to operating away from the design point: a heavier load, a different speed, a thin oil, heat, or misalignment.

This tool follows Chapter 8 of the Naval Surface Warfare Center Handbook of Reliability Prediction Procedures for Mechanical Equipment (NSWC-11). It starts from a base failure rate. That is the manufacturer’s figure, or one failure per 108 revolutions. It then multiplies by six factors: speed, load, misalignment, lubricant viscosity, temperature and the AGMA service factor. A second mode handles splines, where the base rate comes from the spline’s torque, size, hardness and misalignment.

The result is a failure rate in failures per million hours, with FIT and MTBF for a FMECA. Every step of the working is under “Show detailed calculation steps”.

NSWC-11 Chapter 8 Failure Rate Calculator
Gear & Spline Reliability (NSWC-11 Eq. 8-2 / 8-13)
What are you analysing?
Gears use a base rate from the manufacturer or a design life. Splines get their base rate from the Canterbury–Lowther life equation.
Speed & Alignment
The speed the gear or spline was designed and rated for.
Angular misalignment of the mating axes under load. The model has a factor of 1.0 at 0.006 rad (0.34°).
Gear Loading & Base Rate
Use the same unit for both (lb, lb·in of torque, kW, and so on). Only the ratio matters.
The handbook uses 108 revolutions when nothing better is known.
Lubricant & Temperature
Same unit as νO. Only the ratio matters.
The factor starts above 160 °F (71 °C).
AGMA Service Factor (Table 8-1)
Mission (optional)
8,760 h is one year of continuous operation. Leave blank to skip.
Failure Rate & Reliability Metrics
Predicted Failure Rate, λ (failures / 106 h) iλG = λG,B × CGS × CGP × CGA × CGL × CGT × CGV for a gear mesh. A spline uses λGS = λGS,B × CGS × CGL × CGT × CGV.
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Failure Rate (FIT) iFailures In Time = failures per 109 hours. The same number as failures per million hours, multiplied by 1000.
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MTBF (hours) iMean time between failures = 106 / λ. It assumes a constant failure rate. A gear is a wear-out part, so use it to compare designs and to feed a FMECA, not as a life guarantee.
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MTBF (years) iMTBF in hours divided by 24 × 365.25.
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Expected Failures over the Mission iλ × mission hours / 106. The expected count of failures for one unit, so a value above 1 means one failure is more likely than not.
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Survival Probability over the Mission iR(t) = e−λt, the chance one unit survives the mission with no failure, assuming a constant failure rate.
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Intermediates
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Multiplying Factors
SymbolMeaningValue
Show detailed calculation steps
How the Model Works

Gears are built to a specification, and AGMA standards let gears from different makers be compared. NSWC-11 uses that. It takes the gear’s failure rate under its design conditions and adjusts it for how far the real operating conditions differ.

1. The gear equation

\[ \lambda_G = \lambda_{G,B}\cdot C_{GS}\cdot C_{GP}\cdot C_{GA}\cdot C_{GL}\cdot C_{GT}\cdot C_{GV} \qquad\text{(Eq. 8-2)} \]
SymbolMeaning
λGGear failure rate in this operation, failures per million operating hours
λG,BBase failure rate, from the manufacturer or from the design life
CGSSpeed, relative to the design speed
CGPGear loading, relative to the design load
CGAMisalignment
CGLLubricant viscosity, relative to the specification lubricant
CGTOperating temperature
CGVAGMA service factor for the shock and vibration of the application

2. Where the base rate comes from

The best base rate is the manufacturer’s own, quoted at a stated speed, load, lubricant and temperature. Often it is not available. The handbook then notes that a gear or spline is usually designed for a life of 100 million revolutions. So it uses one failure per 108 revolutions:

\[ \lambda_{G,B} = \frac{60\,n}{\text{design life (rev)}}\ \text{per hour} \] \[ \lambda_{G,B} = \frac{60\,n}{\text{design life}}\times 10^6\ \text{per }10^6\text{ h} \]

The page uses the design speed VD for n, because the design life is quoted at that speed. CGS then corrects for running at some other speed. The handbook writes only “RPM”, so this is an interpretation. At 1,800 rpm and 108 revolutions the base rate is 1,080 failures per million hours. That is why gears can look pessimistic next to the other NSWC-11 parts.

3. Splines

A spline has no manufacturer’s rating to lean on. The handbook uses an analytical life from Canterbury and Lowther (reference 11) instead. It depends on the torque, size, hardness and misalignment of the spline. The handbook turns that life into a rate per million revolutions. Speed, lubricant, temperature and service factor are then applied as for a gear. Load and misalignment are already inside the life, so splines do not take CGP or CGA.

4. Step by step

  1. Pick gear or spline.
  2. Find the base rate (gear), or the spline life and its rate (spline).
  3. Multiply by each factor. Each is a power law of how far you are from the design point.
  4. Convert to FIT and MTBF, and, with a mission time, to expected failures and survival probability.
Read the result with care. The factors are steep power laws. The load factor goes as the 4.69 power and misalignment as the 2.36 power. Small input changes move the answer a lot. A gear is a wear-out part, so a constant failure rate describes it roughly. The handbook warns that not every equation could be validated.
Factor Reference Guide
NSWC-11 Chapter 8 — Gears and Splines
λB

Base Failure Rate

Design life or manufacturer value
Base rate \[ \lambda_{G,B} = \frac{60\,V_D}{L_{\text{design}}}\times 10^{6}\ \ \text{failures}/10^6\,\text{h} \]

With the default design life of 108 revolutions, the base rate in failures per million hours is 0.6 × VD. A gear designed for a longer life has a lower base rate in proportion.

Check yourself. VD = 1800 rpm gives 60 × 1800 / 108 = 1.08×10−3 per hour = 1,080 per million hours.
Manufacturer value

If the gear maker gives a failure rate, use it. It is already tied to a stated speed, load, lubricant and temperature, which are the “design” conditions the factors compare against. Choose “Manufacturer’s failure rate” in the form and type it in per million hours.

CGS

Speed Factor: CGS

Equation: NSWC-11 Eq. 8-3, Figure 8.6
Eq. 8-3 \[ C_{GS} = 1.0 + \left(\frac{V_O}{V_D}\right)^{0.7} \]

The oil film between the teeth depends on speed to roughly the 0.7 power (Eq. 8-1). Running faster than design thins the margin, running slower helps. VO is the operating speed and VD the design speed, both in rpm.

As printed, the constant is 1.0, so the factor is 2.0 at the design speed and 1.0 only at standstill. That is how the handbook gives it, and the calculator follows it.
CGS vs speed ratio (NSWC-11 Fig. 8.6)
CGP

Load Factor: CGP

Equations: NSWC-11 Eq. 8-4 to 8-6, Figure 8.7
Eq. 8-6 \[ C_{GP} = \left(\frac{L_O/L_D}{0.5}\right)^{4.69},\qquad 4.69 = 4.56 + 0.13 \]

Load acts two ways. It thins the lubricant film, which goes as the 0.13 power of load (Eq. 8-4). It also drives tooth fatigue, which goes as the 4.56 power (Eq. 8-5). Combined, the failure rate goes as the 4.69 power of the load ratio. The factor is 1.0 at half the design load, so a gear run at its full design load has a factor of 25.8.

LO is the operating load and LD the design load, in any one unit.

The printed figure covers LO/LD from 0.1 to 1.2. The calculator warns above 1.2.
CGP vs load ratio (NSWC-11 Fig. 8.7)
Log scale.
CGA

Misalignment Factor: CGA

Equation: NSWC-11 Eq. 8-7, Figure 8.8
Eq. 8-7 \[ C_{GA} = \left(\frac{A_E}{0.006}\right)^{2.36},\quad A_E\ \text{in radians} \] \[ C_{GA} \approx 12.44\,A_E^{2.36},\quad A_E\ \text{in degrees (Fig. 8.8)} \]

Misalignment of gears, bearings and shafts concentrates load at one end of the tooth and raises vibration. The factor is 1.0 at 0.006 rad (0.344°). The two printed forms agree: 0.006 rad is 0.344° and (0.01745/0.006)2.36 = 12.43.

A power law falls to zero as the angle falls to zero, so a nearly perfect mesh would show a vanishingly small rate. Use a realistic tolerance for the assembly. The calculator warns below 0.03°.
CGA vs misalignment (NSWC-11 Fig. 8.8)
Log scale, angle in degrees.
CGL

Lubricant Factor: CGL

Equation: NSWC-11 Eq. 8-8, Figure 8.9
Eq. 8-8 \[ C_{GL} = \left(\frac{\nu_O}{\nu_L}\right)^{0.54} \]

It has the same form as the bearing lubricant factor. The 0.54 comes from the film-thickness relation, Eq. 8-1. There the film thickness goes as the 0.54 power of the viscosity parameter. νO is the specification lubricant’s viscosity and νL is the viscosity of the lubricant used, both at operating temperature. Thinner oil than specified raises the factor above 1. Only the ratio matters.

CGL vs νO/νL (NSWC-11 Fig. 8.9)
CGT

Temperature Factor: CGT

Equation: NSWC-11 Eq. 8-9, Figure 8.10
Eq. 8-9 \[ C_{GT} = \begin{cases} 1.0 & T \le 160\,^\circ\text{F} \\[4pt] \dfrac{460 + T}{620} & T > 160\,^\circ\text{F} \end{cases} \]

Heat thins the lubricant, relaxes tolerances through thermal growth, and lowers the strength of the gear material. Up to 160 °F the handbook applies no penalty. Above that the factor rises in a straight line with absolute temperature (460 + T is the Rankine temperature), reaching 1.87 at 700 °F.

The equation uses °F as printed. If you enter °C, the calculator converts first. The shaft chapter uses the same equation.
CGT vs operating temperature (NSWC-11 Fig. 8.10)
CGV

AGMA Service Factor: CGV

Table 8-1: Typical AGMA Service Factor

AGMA service factors account for the shock and vibration a gearbox will see. They let you pick a gearbox to match the application. The handbook uses the service factor directly as a multiplier (Eq. 8-10). The table is for a speed-decreasing drive. The row is the character of the prime mover and the column is the character of the load on the driven member.

Prime moverUniform loadMedium shockHeavy shock
Uniform1.001.251.75
Medium shock1.251.502.00
Heavy shock1.501.752.25
The handbook numbers this table 8-1, the same as the gear failure-mode table in section 8.3.
Using your own value

Most manufacturers publish a service factor for each of their products. If you have one, choose “Custom CGV value” for the prime mover and enter it. Example prime movers: an electric motor or steady turbine is uniform, a multi-cylinder engine is medium shock, and a single-cylinder engine is heavy shock. These examples are common AGMA usage and are not printed in the NSWC-11 table.

T 8-12

Spline Life: Canterbury & Lowther

Equations 8-11 to 8-13
Eq. 8-11 – 8-13 \[ \lambda_{GS,B} = \frac{10^{6}}{\theta}\ \ \text{per }10^6\text{ rev} \] \[ \lambda_{GS} = \lambda_{GS,B}\, C_{GS}\, C_{GL}\, C_{GT}\, C_{GV} \] \[ \theta = 7.08\times10^{-10}\left(\frac{\phi\,G_L}{G_D}\right)^{4.56} A_E^{-2.36} \] \[ \phi = \frac{4.85\times10^{3}\,G_B\,G_D^{\,3}}{G_T} \]

θ is the spline life in revolutions. GL and GD are the spline length and diameter (in), AE the misalignment (rad), GT the torque (lb·in) and GB the tooth hardness. Life rises with the 4.56 power of hardness and diameter-cubed over torque, and falls with misalignment to the 2.36 power. Doubling the torque cuts the life by a factor of about 24.

The base rate is per million revolutions. The calculator multiplies by 60 VO revolutions per hour to get failures per million hours.

About the hardness unit
The handbook lists GB as “Tooth hardness (Brinell), lbs/in²”. A typical Brinell value in psi is in the hundreds of thousands. That raises φ so far that the predicted life is astronomically long, around 1026 revolutions or more. With the Brinell number (for example 300) the lives come out in a believable range, so the calculator takes the HB number. If a design is critical, check this against Canterbury and Lowther (reference 11).

The same caution applies to the load factor. The equation is dimensionless only if GB·GD3 and the torque GT are in compatible units. That is another reason to treat spline results as a ranking tool.

The handbook also notes that the most common spline problem is fretting wear, especially in loose splines. Keep the splines flooded with oil, crown them, and confirm that the bearing stress is below the allowable limit.

T 8-1

Gear Failure Modes

The common modes of gear and spline failure are wear, surface fatigue, plastic flow and breakage. In the shear mode a gear stops transmitting power at once. In the wear mode it degrades gradually.

Failure modeFailure causeFailure effect
PittingCyclic contact stress transmitted through the lubrication filmTooth surface damage
Root fillet cracking; tooth end cracksTooth bending fatigueSurface contact fatigue and tooth failure
Tooth shearFractureTooth failure
ScuffingLubrication breakdownWear and eventual tooth failure
Plastic deformationLoading and surface yieldingSurface damage resulting in vibration, noise and eventual failure
SpallingFatigueMating surface deterioration, welding, galling, eventual tooth failure
Tooth bending fatigueSurface contact fatigueTooth failure
Contact fatigueIncorrect heat treatmentTooth failure
Thermal fatigue; abrasive wearContaminants in the gear mesh area or lubrication systemTooth scoring, eventual gear vibration, noise
Condensed from the handbook’s gear failure-mode table. The extracted original is a merged table. The rows here pair the entries as the handbook lays them out.
Worked Example

Gear. A spur gear mesh is designed for 1,800 rpm and a 500 lb tooth load. It runs at 1,500 rpm with a 300 lb load, which is 60 % of design. Misalignment under load is 0.2°. The oil is a 68 grade but the oil in use measures 55 at operating temperature, which is 180 °F. The prime mover is uniform and the driven load is medium shock. No manufacturer rate is available, so the 108-revolution default is used. These are the calculator’s default inputs.

Step-by-step (gear)
  • λG,B = 60 × 1800 / 108 × 106 = 1,080 per 106 h
  • CGS = 1 + (1500/1800)0.7 = 1.8802
  • CGP = ((300/500)/0.5)4.69 = 1.24.69 = 2.3516
  • CGA = (0.2° = 0.003491 rad / 0.006)2.36 = 0.2785
  • CGL = (68/55)0.54 = 1.1214
  • CGT = (460 + 180)/620 = 1.0323
  • CGV = 1.25 (uniform prime mover, medium shock load)

λG = 1080 × 1.8802 × 2.3516 × 0.2785 × 1.1214 × 1.0323 × 1.25 ≈ 1,924 failures / 106 h, an MTBF of 520 h. The combined multiplier is 1.78. Load is the factor to look at: the gear is at 60 % of design load, and the factor is already 2.35.

Spline. Switch to spline mode. A spline is 2 in long and 2 in in diameter, with 300 HB hardness. It carries 20,000 lb·in. Speed, oil, temperature, service condition and the 0.2° misalignment are the same as before.

  • φ = 4850 × 300 × 23 / 20000 = 582
  • θ = 7.08×10−10 × 5824.56 × 0.003491−2.36 = 1.806×109 revolutions (20,071 h at 1,500 rpm)
  • λGS,B = 106/θ = 0.5536 per 106 rev, which is 0.5536 × 60 × 1500 = 49.82 per 106 h
  • Factors: CGS = 1.8802, CGL = 1.1214, CGT = 1.0323, CGV = 1.25, a product of 2.7206

λGS = 49.82 × 2.7206 ≈ 135.5 failures / 106 h, an MTBF of 7,378 h. Try 30,000 lb·in: the life drops to 3,159 h, because torque enters as the 4.56 power.

Important Notices
  • Not an official DoD document. NSWC-11 is the product of a Naval Surface Warfare Center research program, approved for public release. The handbook cautions that limited funding prevented full validation of every prediction equation. It should not be treated as an official Department of Defense standard.
  • No Navy affiliation or endorsement. The Naval Surface Warfare Center, Carderock Division and the U.S. Navy have not participated in the development of this calculator and do not approve or endorse it.
  • Use with the full procedure. NSWC-11 warns against extracting equations without regard to application procedures and parameter limits. Results are a design screening tool, not a substitute for the manufacturer’s rating, testing, or the judgment of a qualified engineer.
  • Gears and splines only. The model covers the gear mesh and splines. The bearings and shafts in a gearbox are separate parts. The handbook points out that bearings are usually the main driver of gear-system reliability. Use the bearing and shaft calculators for those.
  • Design conditions matter. The load, speed and viscosity factors are ratios against the design conditions. They are only meaningful if the design values you enter are the ones the gear was actually rated at.
  • Constant failure rate. An MTBF assumes a constant failure rate. Gear wear and fatigue are wear-out mechanisms, so treat the estimate as a rough guide over long missions.
  • Spline hardness unit. See the spline card for the unit question. If the result matters, compare it with the original Canterbury and Lowther report.

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